Advances in Pure Mathematics, 2013, 3, 178-182
http://dx.doi.org/10.4236/apm.2013.31A025 Published Online January 2013 (http://www.scirp.org/journal/apm)
Cyclic Operator Decomposition for Solving the
Differential Equations
Ivan Gonoskov
Institute of Applied Physics of the Russian Academy of Sciences, Nizhny Novgorod, Russia
Received October 19, 2012; revised November 23, 2012; accepted December 6, 2012
ABSTRACT
We present an approach how to obtain solutions of arbitrary linear operator equation for unknown functions. The par-
ticular solution can be represented by the infinite operator series (Cyclic Operator Decomposition), which acts the gen-
erating function. The method allows us to choose the cyclic operators and corresponding generating function selectively,
depending on initial problem for analytical or numerical study. Our approach includes, as a particular case, the pertur-
bation theory, but generally does not require inside any small parameters and unperturbed solutions. We demonstrate
the applicability of the method to the analysis of several differential equations in mathematical physics, namely, classi-
cal oscillator, Schrödinger equation, and wave equation in dispersive medium.
Keywords: Operator Decomposition; Spectral Theory; Propagator
1. Introduction
Various classical and quantum-mechanical problems in
theoretical physics lead to the necessity of solving the
linear operator equations for unknown functions and, in
particular, the differential equations. Exact non-trivial
analytical solutions of these equations, which include
finite combinations of elementary operations and special
functions, are known only for a number of specific cases.
However, there are many actual and important cases for
which such exact solutions were not still obtained even
by using severe approximations for the corresponding
interaction operators. For the cases when exact solutions
are unknown, some approximate methods are usually
used. They can be conventionally divided into two types:
1) varieties of perturbation theory and 2) numerical cal-
culations (which generally are also based on perturbation
theory). In spite of significant usefulness and applicabil-
ity, these methods are not free from various limitations
and disadvantages. The perturbation theory approaches
may lead to divergent series, they need sometimes suit-
able unperturbed solutions, and, finally, they do not pro-
vide even estimations for precision in most cases (see
[1,2] and references therein). On the other hand, numeri-
cal schemes, which are from the very beginning ap-
proximate, usually also do not give reliable estimations
for the precision (some reasonings can be found in [3,4]).
Moreover, they can hardly give an asymptotic behavior
of the solutions at infinity. Thus, the development of the
general method which allows to overcome some of the
above-mentioned difficulties is the main object of our
study.
In this manuscript we develop an approach based on
the theory of Cyclic Operator Decomposition (COD),
which gives the opportunities to obtain solutions (exact
or approximate) of the differential equations with arbi-
trary operators. The particular solution can be repre-
sented by the infinite cyclic operators series, which acts
the previously determined generating function. The cy-
clic operators and the corresponding generating function
(COD components) can be specified through the given
operators in the differential equation. Under the conver-
gence requirement, these COD components can be cho-
sen in different ways depending on the certain problem
statement. The procedure differs from the using of Born
series (or corresponding Neumann series) in the pertur-
bation theory [5-8] and S-matrix theory of Heisenberg,
Feynmann and Dyson [9]. It can be understood easily by
studying, for example, the difference between the formal
definition of the generating function and Green’s func-
tion (the last one is derived in some cases by using the
operator resolvent formalism) [5,7,10-12]. Generally, the
proposed series does not require any small parameters or
unperturbed solutions for the convergence. But, as a
matter of fact, the procedure can be transformed, under
some certain choice of COD components, to the “stan-
dard” perturbation theory with small parameters. For the
potentials without strong singularities, with reasonable
C
opyright © 2013 SciRes. APM

I. GONOSKOV 179
choice of the cyclic operators and generating function,
the corresponding series usually has uniform conver-
gence. Some additional features and advantages of our
approach for analytical and numerical solving the differ-
ential equations are demonstrated in sections below.
2. Theory of Cyclic Operator Decomposition
Let us start from the general case of operator equation for
unknown function:
ˆ0.D
 
ˆ
D
(1)
Here is an arbitrary given linear operator and

is an unknown function, which can be a vector or matrix
of arbitrary dimensionality. This equation can lead in
particular cases to arbitrary linear differential equations,
which are considered in examples below.
Let us consider a pair of operators and V, which
are determined by the following condition:
ˆ
Gˆ
ˆ ˆˆ
.DG V
ˆ
G
11
ˆˆ
ˆˆ
at: ;GGGI


ˆ
hat: 0,
gg
G   
ˆ
(2)
Since the choice of this pair is partly optional, we
impose additional conditions on the operator :
,t h (3)
0, t
 
 (4)
where
I
is the identity operator. Any function
g

,
which satisfies Equation (4), will be called generating
function. Now we can write the following equation:

ˆ

1
ˆˆ.
g
IGV



ˆ
G
(5)
As we can check, under the above-mentioned condi-
tions for and
g

, any solution of Equation (5) ful-
fills Equation (1). Equation (5) can be solved in terms of
the following Cyclic Operator Decomposition:

11
1
1
ˆ ˆˆˆ
ˆ ˆˆ
1
ˆˆ
ˆ
.
g
n
n
IGVGV
IGV
g
GV







1
ˆˆ
GV





 





 (6)
This is the exact particular solution of Equation (1)
with corresponding particular COD components deter-
mined by Equations (2) and (3). The solution makes
sense only if the obtained series is convergent. This can
be achieved in different cases depending on and
g

, for example, if we work in Banach space and corre-
sponding operator norm is 1
ˆˆ1GV

ˆ,D
. The convergence
of some similar operator series was considered also in
[5,13].
The theory can be easily generalized also to the case of
the equations with given sources:



(7)
where an arbitrary given function

describes the arbi-
trary sources. The unknown function

could be found
naturally if the inverse operator is known: 1
ˆ
D




1
ˆ
D
 
11
1
ˆˆ ˆˆn
g
n
IGV G
.
However, the inverse operator can not be easily
found for a number of problems. Then, we can write a
solution of this equation analogously by using COD:
.




 



0
g
(8)
In contrast to the case of Equation (1), now we can
choose

for some non-trivial particular solutions.

Then we can obtain the particular solution of Equation
(7), which corresponds to the following particular deter-
mination of the inverse operator in terms of COD:

111
1
ˆˆ ˆˆ
.
n
n
DI GVG







ˆ
G

(9)
Important feature of the proposed theory is that, while
the conditions Equations (3) and (4) should be fulfilled
and the convergence of the series is necessary, we still
have a great freedom of choosing and corresponding
. Generally, it gives us opportunities to obtain all the
g

possible solutions of Equation (1). Sometimes we can
naturally choose and
ˆ
G
g

in accordance, for exam-
ple, with the corresponding initial conditions for Cauchy
problem or boundary conditions for boundary-value
problems.
In some cases, the exact solution Equation (6) can be
used naturally for obtaining the approximate solution
with finite number of terms. It can be done, for example,
when, starting from certain number n, the following con-


1
11
ˆˆ ˆˆ
nn
gg
GV GV 


 ditions are satisfied: .
These are the sufficient conditions enabling one to derive
the approximate solution with the prescribed accuracy.
Further, the proposed method provides another advantage
if one performs numerical calculations. According to the
exact solution Equation (6), we can use recurrent rela-
tions when calculating numerically the approximate solu-
tions. In this case, the calculation of any next term in the
corresponding series does not require more numerical
resources than the calculation of the previous one.
3. Examples
In this section we apply the proposed theory of Cyclic
Operator Decomposition for the various cases of differ-
ential equations. Let us first consider the Cauchy prob-
lem for the equation of classical oscillator. Note that this
equation, if written in other variables, is the stationary
one-dimensional Schrödinger equation with given energy,
and it can be transformed also to the Riccati equation by
using logarithmic substitution.

I. GONOSKOV
180



20,
,
,
a
b
ftf
a
b
 



ft
ft


(10)
where
f
t

2t
 is an unknown function, is an ar-
bitrary time-dependent frequency and a, b are arbitrary
constants. Here, it is natural to choose the components
for COD as follows:


12
,,
ˆ
, .
ga
btt
GVt 




dd
d
aa
a
1
2
2
12
d
ˆ
d
ˆdd
ab
t
tt
Ga
t







(11)
If we fix (by our local convention, which we will use
below) that we write for brevity the same variable upper
limit of integration as the integration variable and deter-
mine the successive integration (step by step from right
to left), we can write a simple expression for the solution:
  
 
2
22
dd d
ab
ab ab
tt
tt
tt tt
tt tt
f
tabttt ta
ttttta


   
 


 
bttt
bttt
 





2t


(12)
It is important to note now, that the presented series
(Equation (12)) has rapid uniform convergence at least in
any interval, where is bounded. For example, in
the limited interval

0,t

the rate of convergence for
f
t
0
can be estimated in the following way (we assume
here for simplicity, that ab
tt

0b, , and the
maximum of

2t
 in the corresponding interval is
):
max
C



2
max
2!
nn
Ct
na
n
-thtermofseries. (13)
In the same way, the convergence can be demonstrated
for other different COD’s, when the cyclic operators are
bounded for the given generating functions in the given
relevant interval. Moreover, if additionally


t 
 
22
 0t
is a
real function, and , we have a
decreasing alternating series for the above example, and
we can estimate the precision of partial sum of the series
by the value of the last term.

t00
Now we focus on some particular cases of


2t
 . To
demonstrate that it is possible to obtain a solution with
any prescribed precision, we consider a case when

2

1
1sin 0
2
tt

 


1a0b

and , , . Calculation of the first
two terms in Equation (12) gives
0
ab
tt

2
1
1sin
2
ftt tt.



 


0.0273
(14)
By calculating the third term in Equation (12), we
obtain

if we consider t in the interval [0,1].

Sometimes we can find also the asymptotic behavior
of the solution. As an example, we consider the case


2tt

0 and ab
tt1a0b, , , where




is
an arbitrary constant, 1


  
. Using again Equation (12)
we obtain exact solution in the following form:
 
2
24
112
122324
t
ft
t



 


  (15)


 


0t
  
By analyzing the corresponding series we can obtain a
simple upper estimate for the solution :
11
22
2
1exp,
12 2
tt
ft


 



 
 


:t

(16)
which gives us the following asymptotic behavior at
11
2
2
exp .
2
t
ft


 





1
(17)
In this case, the same asymptotic can be found also
from WKB theory (see, for example, quasiclassical ap-
proximation in [14,15]).
Let us now demonstrate the selective choice of cyclic
operators. For that we consider stationary one-dimen-
sional Schrödinger equation (we use below the units
where

1
p
m
,

):

2
2
d
2e0,
d
x
EA x
x




 


(18)


x
where

2
2Em1
is an unknown function, which describes
quantum state with energy E in the continuum; A, β are
arbitrary real constants. Without loss of generality (one
can use scale transformations of Equation (18)) we can
assume ,


. To find the solution of this
equation, we can choose the components for COD in
different ways. For example, if one interests in the be-
havior of


x
 0x near

and in small values of E,
he can choose
2
2
d
ˆ
d
G
x

1
ˆ
G
and use nearly the same tech-
nique as in Equations (11) and (12). However, this choice
can be inconvenient for the analysis of the long-range
behavior. Another variant of choosing the components
for COD is the following (we use also Equation (9) for
the particular determination of ):

I. GONOSKOV 181


2
2
2
11
000
1
12
00
d
ˆ,e
d
ˆˆˆˆ ˆ
1
ˆdd,
g
k
k
xx
Gmx C
x
GGV
GxxV





 
 








12
1
e,
,
ˆˆ
, e,
imx imx
x
C
G
mVA





eimx
gx
 


p
(19)
where C1 and C2 are arbitrary constants. To obtain the
general solution, we consider the particular case of gen-
erating function and corresponding par-
ticular solution
x
 . Then we derive by using a rule
of infinite geometric series:









1
2
1
22
e
1
e
11
im x
kimx
A
im
mA
im











11
00
1
2
1
1
ˆˆ
ˆ
ˆˆ
1
1
e.
12
k
g
K
K
im x
GV xGV
im
A
im














 







(20)
From here we obtain

2
1
2ki
mk





2pp
11
e1 e
n
imxn nx
p
nk
xA 



 


 (21)
and the general solution in the following form:
 
1
x
Cx   Cx 

 (22)
The corresponding series converges at any A and real
m.
In a similar way we can obtain solutions for multi-
dimensional equations. Let us consider the stationary
Schrödinger equation with potential surface 

Ur, m-
dimensional Laplace operator ∆, and energy E:
ulti
  
20. 



rr EU

 
 (23)
Then we can choose

 
,2
EUr
ˆˆ
GV , and
g

is any solution of 0
g

. From corresponding
COD we can obtain a solution:

11
1VV 


 

r10.
g
V 

 


1 2
ˆ
(24)
The inverse Laplace operator can be written, for ex-
ample, as 1
ˆ
F
kF

ˆ , where
F
is the Fourier
transform operator and k is an absolute value of the wave
vector in this transform.
Another choice of COD components can be better for
the finding of the bound states with . We can 0E

2Ur choose: ,
ˆ
GEˆ 2,V
g

is any solution of


2
g
E   0, and write the inverse operator 1
ˆ
G

for
COD as follows:
1
11
2
1
ˆˆˆ
2.
2
GE FF
Ek


 (25)
In this way, we can calculate in some cases the terms
in the corresponding COD by evaluating the poles at
imaginary values 2
P
kiE .
Now we consider time-dependent three-dimensional
Schrödinger equation to demonstrate other applications
of the proposed method. Let us consider propagation of
charged particle with arbitrary electromagnetic interactions
(below


,t
A
r1qc is the vector potential, and ):

  
2 1,,,0.
2
ii tUtt
t




 



Arr r (26)
Here, we can choose the components for COD in a
variety of ways depending on peculiar properties of the
interactions. One special choice is the following:

  
0
00
1
2
ˆ,(,) ,
ˆd,
1
ˆ,,,
2
g
t
t
Gi t
t
Git
Vi tUt
 


 


 

rr
Ar r
ˆ
V
 
000
2
0
ˆˆˆ
,1 ddd.
ttt
ttt
titVitVtV 






rr
ˆ
P
(27)
where corresponds to the time-dependent Hamilto-
nian. It gives the following solution:
(28)
This solution can be useful for the numerical calcula-
tions, namely, for the finding the propagator , which
gives:





ˆ
,,
n
ttP tOt   rr

ˆ,tr

,t
, see also [4] and
references therein. If we use additionally internal time-
ordering, we can transform this expression to Dyson se-
ries (see [9,16]).
Finally, we consider the wave equation for electro-
magnetic waves in dispersive medium. We assume that
an arbitrarily given operator  which describes
electric dispersion does not depend nonlinearly on the
field, i.e. we still have linear problem. In this case the
equation for unknown vector potential
A
r is the
following (see for example [17]):
 
2
ˆ,,0, tt
tt
 

 





rAr


(29)
with the initial conditions


0
0
,.
t
t
t








A
A
SrRr (30)
We can choose the following components for COD:

I. GONOSKOV
182



0
11
ˆˆ
ˆ
ˆˆ
d,
t
g
t
Gt
tt
Gt









r

00
1
2
,,
d, ,
ˆ
d,
tt
tt
tt
ttV




S
r
r



rRr


0
12
,d
d, .
n
tt
t
tt
tt



 









r
r Rr

 

(31)
Then, we can find the solution, which follows from the
corresponding COD series:


0
0
1
1
ˆ
,1 d
ˆ
nt
t
t
tt






 









Ar
Sr


(32)
For this solution the initial magnetic field is equal to
S
r

Rr
and the initial electric field is equal to
.
4. Conclusion
In summary, we propose the theory of Cyclic Operator
Decomposition, which allows one to obtain particular
solutions of linear operator equations for unknown func-
tions. In most cases it is possible to obtain all the possi-
ble solutions, which satisfy the given conditions. We
demonstrate by some reasonings and particular examples
that our approach has the following remarkable proper-
ties: 1) there is a freedom in choosing the COD compo-
nents depending on the certain problem; 2) there is a
rapid uniform convergence for most of the considered
cases; 3) it is possible to find the asymptotic behavior of
the solutions; 4) in many cases when one is analyzing the
approximate solution, it is possible to estimate the accu-
racy; 5) the proposed approach gives good opportunities
for efficient implementation of numerical calculations
due to the recurrent relations that can be used in COD.
5. Acknowledgements
Author would like to thank academician L. D. Faddeev,
M. Yu. Emelin, M. Yu. Ryabikin, and A. A. Gonoskov
for the useful and stimulating discussions.
REFERENCES
[1] F. J. Dyson, “Divergence of Perturbation Theory in Quan-
tum Electrodynamics,” Physical Review, Vol. 85, No. 4,
1952, pp. 631-632. doi:10.1103/PhysRev.85.631
[2] A. V. Turbiner, “The Eigenvalue Spectrum in Quantum Me-
chanics and the Nonlinearization Procedure,” Soviet Phy-
sics Uspekhi, Vol. 27, No. 9, 1984, p. 668.
doi:10.1070/PU1984v027n09ABEH004155
[3] J. Fleck, J. Morris and M. Feit, “Time-Dependent Propa-
gation of High Energy Laser Beams through the Atmos-
phere,” Applied Physics A: Materials Science and Proc-
essing, Vol. 10, No. 2, 1976, pp. 129-160.
doi:10.1007/BF00896333
[4] A. N. Drozdov and S. Hayashi, “Numerical Test of Ap-
proximate Single-Step Propagators: Harmonic Power Se-
ries Expansions versus System-Specific Split Operator Re-
presentations,” Physical Review E, Vol. 59, No. 2, 1999,
pp. 1386-1397. doi:10.1103/PhysRevE.59.1386
[5] J. V. Corbett and J. Math, “Convergence of the Born Se-
ries,” Journal of Mathematical Physics, Vol. 9, No. 6,
1968, p. 891. doi:10.1063/1.1664655
[6] M. Wellner, “Improvement of the Born Series at Low En-
ergy,” Physical Review, Vol. 132, No. 4, 1963, pp. 1848-
1853. doi:10.1103/PhysRev.132.1848
[7] R. Perez and J. Math, “On the Expansion of the Propaga-
tor in Power Series of the Coupling Constant,” Journal of
Mathematical Physics, Vol. 20, No. 2, 1979, p. 241.
doi:10.1063/1.524070
[8] F. S. Bemfica and H. O. Girotti, “Born Series and Unitar-
ity in Noncommutative Quantum Mechanicsphys,” Physi-
cal Review D, Vol. 77, No. 2, 2008, Article ID: 027704.
doi:10.1103/PhysRevD.77.027704
[9] F. J. Dyson, “The S Matrix in Quantum Electrodynam-
ics,” Physical Review, Vol. 75, No. 11, 1949, pp. 1736-
1755. doi:10.1103/PhysRev.75.1736
[10] J. H. Eberly and H. R. Reiss, “Electron Self-Energy in In-
tense Plane-Wave Field,” Physical Review, Vol. 145, No.
4, 1966, pp. 1035-1040. doi:10.1103/PhysRev.145.1035
[11] H. R. Reiss and J. H. Eberly, “Green’s Function in In-
tense-Field Electrodynamics,” Physical Review, Vol. 151,
No. 4, 1966, pp. 1058-1066.
doi:10.1103/PhysRev.151.1058
[12] A. Mockel and J. Math, “Invariant Imbedding as a Gener-
alization of the Resolvent Equation,” Journal of Mathe-
matical Physics, Vol. 8, No. 12, 1967, p. 2318.
doi:10.1063/1.1705158
[13] B. Ćurgus and T. T. Read, “Discreteness of the Spectrum
of Second-Order Differential Operators and Associated
Embedding Theorems,” Journal of Differential Equations,
Vol. 184, No. 2, 2002, pp. 526-548.
doi:10.1006/jdeq.2001.4152
[14] L. Landau and E. Lifshitz, “Quantum Mechanics Non-Re-
lativistic Theory,” 3rd Edition, Pergamon Press, Oxford,
1977.
[15] L. Schiff, “Quantum Mechanics,” 3rd Edition, McGraw
Hill, New York, 1968.
[16] F. J. Dyson, “The Schrödinger Equation in Quantum Elec-
trodynamics,” Physical Review, Vol. 83, No. 6, 1951, pp.
1207-1216.
[17] L. Landau and E. Lifshitz, “Electrodynamics of Continu-
ous Media,” Vol. 8, Butterworth-Heinemann, Oxford, 1984.